- Zeno's Paradoxes
- First off, let's get some distance from the particular paradoxes.
- What is Zeno trying to do generally?
- Prove that Parmenides is right: everything is unified: there is no
movement, no time, no alteration
- Prove that reality is one thing, unchanging, and indivisible.
- There is no part that is more or less real nor is there more than
one reality or truth or being
- As an aside, there are those who think that quantum general
relativity is on Parmenides' side: they know a lot more than I
about their theories and I cannot pretend to judge their
arguments, but it's interesting.
- Many people criticized Parmenides because what he claimed seemed so
preposterous
- Zeno wants to show that assuming that the world is the way people
claim is also absured
- Claiming that things move has absurd consequences
- Claiming that there are many things has absurd consequences
- etc.
- Digression: Argument form:
- Ad absurdum:
- You claim one thing. You prove it by assuming the contrary of that
claim, just for the sake of argument. Then you show that absurd,
unacceptable, obviously false, or contradictory results follow.
- Some everyday examples, truncated:
- You need light to see, otherwise you could see in the dark.
- You have to brush your teeth, otherwise your teeth will fall
out.
- Ad hominem:
- an argument addressed to a particular position or a particular
person.
- In this case, the argument only really works against that
position or person, but may have huge flaws otherwise.
- I can argue against racism by arguing that Zorgunthol's
particular racist theory is wrong: that is an ad hominem argument.
- This is the sort Zeno uses: he is trying to take people's own
beliefs and defeat them. He is not concerned with whether those
beliefs are right or not.
- Another way to make an ad hominem argument is less
respectable: it is downright false
- Some everyday examples, truncated:
- You know nothing about fashion: just look at your car.
- If your mother is such a good pool player, why does she
miss every shot she takes?
- The argument that what they discharge into the river is
harmless is wrong: that company is a for profit entity.
- who are you to say I stink: have you sniffed your own
shoes?
- Paradox.
- A paradox is what results from taking reasonable assumptions and
deducing a contradiction or unacceptable consequence from them.
- Dilemma.
- A dilemma is an argument in which you claim that there are two
possible ways to go, but both have unacceptable consequences.
- Damned if you do, damned if you don't.
- A catch-22.
- Particular absurdities which Zeno pointed out:
- If you think that motion is infinitely divisible, then you must
believe that nothing moves (the arrow argument).
- If you think there are many things, then Zeno points out that you
must conclude that things are both infinitely large and infinitely
small (the dichotomy argument)
- If there are many, they must be as many as they are and neither
more nor less than that. But if they are as many as they are, they
would be limited. If there are many, things that are are unlimited.
For there are always others between the things that are, and again
others between those, and so the things that are are unlimited. (Simplicius(a)
On Aristotle's Physics, 140.29)
- Claims that claiming there is more than one thing involves
contradiction.
- Think of birds on a telephone wire: 10 of them.
- There is no bird between bird #3 and bird #4. Zeno's argument does
not work.
- But we want to see what he meant, and this is so obvious that he
cannot have meant that!
- There is air between the birds!
- But between bird #1 and the air that touches it, there is no
thing.
- So there is an easy limit: there are 10 birds and 9 chunks of air
between them.
- But we want to see what he meant, and this is so obvious that he
cannot have meant that!
- What of points on a line?
- Between every single pair of points, there is a point half way
between them, so there are always more points!
- This works like Zeno's argument!
- But wait: how are the things in this world points?
- Are things really point-like?
- Well, no. But modern mathematicians are curious about the points
point too, and they think that in fact some infinities are bigger
than others. A man named Cantor worked out in the 19th century a
way to treat infinite numbers as definite, believe it or not, and
so Zeno's paradox may not even apply to points on a line!
- … if it should be added to something else that exists, it would not
make it any bigger. For if it were of no size and was added, it cannot
increase in size. And so it follows immediately that what is added is
nothing. But if when it is subtracted, the other thing is no smaller,
nor is it increased when it is added, clearly the thing being added or
subtracted is nothing. (Simplicius(a) On Aristotle's
Physics,139.9)
But if it exists, each thing must have some size and thickness, and
part of it must be apart from the rest. And the same reasoning holds
concerning the part that is in front. For that too will have size and
part of it will be in front. Now it is the same thing to say this once
and to keep saying it forever. For no such part of it will be last,
nor will there be one part not related to another. Therefore, if there
are many things, they must be both small and large; so small as not to
have size, but so large as to be unlimited. (Simplicius(a) On
Aristotle's Physics, 141.2)
- suppose there are more than one things.
- since he's just argued that things with no spatial extent do not
exist, we must suppose that at least one spatially extended object
exists.
- If spatially extended, it has a front and a back and parts, and
each part has extension too. And each part has parts that have
spatial extension, and each part of a part of a part ...
- If you add up all those parts, you get an infinitely large object.
- Because adding up an infinite number of objects each of which has
spatial extension yields an object of ... infinite spatial
extension!
- Right? No, wrong.
- Because there is a limit.
- 1/2+1/4+1/8+1/16 .... = (wait for it...) 1
- But now come the modern mathematicians:
- What about the series 1-1+1-1+1-1...
- You can rewrite it as (1-1)+(1-1)+(1-1)...
- And in that case, it seems that (1-1)+(1-1)+(1-1)...=0
- But then you can rewriteit as 1+ ((1-1)+(1-1)+(1-1)...)=1
- Seems preposterous!
- Cauchy worked it out and said that such a series is simply
"undefined"!
- And what if Zeno were clever and said that he was not just
dividing one half in half, but rather both halves in half each time?
- Then he would be summing up an infinite number of equally-sized
itty-bitty bits, and that sum does indeed seems to be infinitely
large.
- If we say that he has divided them into not just itty-bitty
bits, but bits so small that they have no extension at all, then
we have another problem, because then they add up to a sum that
has no extension either!
- … whenever a body is by nature divisible through and through,
whether by bisection, or generally by any method whatever, nothing
impossible will have resulted if it has actually been divided …
though perhaps nobody in fact could so divide it.
What then will remain? A magnitude? No: that is impossible, since
then there will be something not divided, whereas ex hypothesi the
body was divisible through and through. But if it be admitted that
neither a body nor a magnitude will remain … the body will either
consist of points (and its constituents will be without magnitude) or
it will be absolutely nothing. If the latter, then it might both
come-to-be out of nothing and exist as a composite of nothing; and
thus presumably the whole body will be nothing but an appearance. But
if it consists of points, it will not possess any magnitude. (Aristotle
On Generation and Corruption, 316a19)
- The first asserts the non-existence of motion on the ground that
that which is in locomotion must arrive at the half-way stage before
it arrives at the goal. (Aristotle Physics, 239b11)
- The [second] argument was called “Achilles,” accordingly, from the
fact that Achilles was taken [as a character] in it, and the argument
says that it is impossible for him to overtake the tortoise when
pursuing it. For in fact it is necessary that what is to overtake
[something], before overtaking [it], first reach the limit from which
what is fleeing set forth. In [the time in] which what is pursuing
arrives at this, what is fleeing will advance a certain interval, even
if it is less than that which what is pursuing advanced … . And in the
time again in which what is pursuing will traverse this [interval]
which what is fleeing advanced, in this time again what is fleeing
will traverse some amount … . And thus in every time in which what is
pursuing will traverse the [interval] which what is fleeing, being
slower, has already advanced, what is fleeing will also advance some
amount. (Simplicius(b) On Aristotle's Physics, 1014.10)